Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is total if any two elements that are related in one order or the other:
In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is total if its union with its reverse is the universal relation:
Of course, this containment is in fact an equality.
A total relation is necessarily reflexive.
Note that an entire relation is sometimes called ‘total’, but these are unrelated concepts. The ‘total’ there is in the sense of a total (as opposed to partial) function, while the ‘total’ here is in the sense of total (as opposed to partial) order.